In this article we report on a study focused on revealing and categorizing the arguments that preservice mathematics teachers put forward when they are asked about why mathematics is taught, which is a question closely related to the justification problem in mathematics education. Another focus of the study is the identification of myths within…

How people see the world, even how they research it, is influenced by beliefs. Some beliefs are conscious and the result of research, or at least amenable to research. Others are largely invisible. They may feel like "common knowledge" (though myth, not knowledge), unrecognized premises that are part of the surrounding culture. As we…

Metacognition is central to children's cognitive development. However, there is conflicting evidence about children's ability to accurately monitor their performance and subsequently control their behavior. This is of particular interest for mathematics topics on which children exhibit persistent misconceptions--that is, when children's knowledge…

Presents research findings related to students' intuitive ideas about the concepts of chance to inform teachers how students form their concepts of probability and statistics. Discusses adolescents' conceptions of uncertainty, judgmental heuristics in making estimates of event likelihood, the conjunction fallacy, the outcome approach, attempts to…

Discusses misconceptions about constructivism by identifying related myths such as students should always be actively and reflectively constructing, manipulatives make learners active, and cooperative learning is constructivist. One central goal of constructivism should be that students become autonomous and self-motivated in their learning.…

Discusses geometrical reasoning in the framework of the theory of Figural Concepts to highlight the interaction between the figural and conceptual components of geometrical concepts. Examples of students' difficulties and errors in geometrical reasoning are interpreted according to the internal tension that appears in figural concepts resulting…

Discusses the use of feedback from students and the analysis of students' error patterns to understand why students develop misconceptions in mathematics. Looks at learning vis-a-vis the abstract nature of mathematics and the students' cognitive development. (MDH)

When we consider the gap between mathematics at elementary and secondary levels, and given the logical nature of mathematics at the latter level, it can be seen as important that the aspects of children's logical development in the upper grades in elementary school be clarified. In this study we focus on the teaching and learning of "division with…

The theme of the material contained in this annotated resource list is the relationship between teaching and learning mathematics in the specific content area of graphing functions. The list contains 30 articles, papers, and unpublished manuscripts written from 1979 through 1989. The articles treat various aspects of concept formation,…

Uses manipulative materials to build and examine geometric models that simulate the self-similarity properties of fractals. Examples are discussed in two dimensions, three dimensions, and the fractal dimension. Discusses how models can be misleading. (Contains 10 references.) (MDH)

Pragmatic and semantic problem solving are examined as processes that enhance acquisition of mathematical knowledge. It is suggested that development of mathematical cognition involves restructuring and that math teachers can help restructure children's knowledge systems by providing them with situations in which semantic and pragmatic problem…

Examined is the situation in which pupils invent mathematics on their own and teachers' reactions to this situation. The assimilation of students' original ideas into correct mathematical concepts is discussed. (CW)

Reviews the logic underlying mathematical induction and presents 10 tasks designed to help students develop a conceptual framework for mathematical induction. (Contains 20 references.) (MDH)

Proposes analogy as the central mechanism of knowledge acquisition in formal domains. Discusses experimental data on preschoolers' knowledge of one-to-one correspondence and college students' understanding of force decomposition. Suggests that a knowledge base domain is a thematically organized knowledge structure and that thematic relations in a…

Three teaching experiments are reported which study aspects of a diagnostic teaching methodology. An experiment in the field of directional quantities showed a positive relationship between the intensity of discussion and amount of learning; one on fractions and another on geometric reflections showed good two-month retention under the…